Well the problem does not go away. I have tried purging all caches, restarting dns servers and clients, and nothing has changed.

Using my iphone as the test client - if I connect to the network via wifi, and go to our Moodle homepage, a page loads as if I am another user - but with almost no functionality (links don't work etc, I get in a cycle of logging in/out).

If I turn off wifi and connect via phone coverage (therefore external connection to our Moodle server) everything is fine.

So - something tells me it is a network issue that is not playing nice with our Moodle setup. Any ideas?

Something either in config of apache server (rewrite rule?) or Moodle OR network (as Dan has also suggested) is injecting (tacking onto the end of them) the '&MoodleSession=blah blah blah' in the URL's.